1998 Changes and additions for the Consequences of the Axiom of Choice Project

  1. Changes May 1998

  2. The final two paragraphs of note 94 occur twice. One occurrence should be deleted.

  3. In problem three of note 0, ZF should be replaced by ZF$^0$.

  4. Changes September 1998

    In Model N19 in the list of forms that are false, replace 153 by 152, then the list of forms that are false in this model are as follows:

    A minimal list of forms at least one of which is implied by each form known to be false in the model:
    15, 51, 59, 118, 144, 192, 152, 253, 328.

    A list of all forms known to be false in the model:
    1, 15, 40, 51, 59, 66, 67, 89, 90, 95, 101, 109, 112, 114, 118, 133, 144, 149, 152, 179, 192, 202, 214, 218, 253, 255, 256, 257, 258, 259, 260, 261, 262, 264, 286, 292, 328, 333, 334, 335, 347, 359, (383).

    In part V, References for relationships between forms, remove the line:

    126 5 (3) Saveev [1975] (M6)

    and in the file rfb1.tex remove the line:

    126 5 (3) Saveev [1975] (M6)
    or get the most recent version of rfb1.tex.

    Then run the program mk_book to correct tables 1 and 2.

    Several forms have been shown equivalent to form 94. For these changes choose one of the two below:

  5. Changes in November 1998

    Some results of De la Cruz and Di Prisco have been added to the project. For these additions choose one of the two below:

    In the bibliography, change Howard/Keremedis/Rubin/Rubin [1997a] and [1997b] to:

    [1998a] Versions of normality and some weak forms of the axiom of choice, Math. Logic Quart. 44, 367-382.
    [1998b] Disjoint unions of topological spaces and choice, Math. Logic Quart. 44, 493-508.

    In the bibliography, change Brunner/Howard/Rubin [1996] to:

    [1997] Choice principles in Weglorz' models, Fund. Math. 154 97-121.

    Further Changes in November 1998

    The descriptions of the models N15, N17 and N36($\beta$) have been revised. For these revisions choose one of the two below:

    M. Higasikawa has brought to our attention the fact that we had not included some results from Higasikawa [1995] in our book. Consequently, add a new form to Part I.
    FORM 75. If a set has at least two elements, then it can be partitioned into well ordered subsets, each of which has at least two elements. Higasikawa[1995].
    (This form should be added to the section ORDERING PRINCIPLES INCLUDING PROPERTIES OF PARTIAL ORDERS, I. Linear And Well Orderings, in PART II.)
    Also, add two lines to note 120.
    57. 20 + 75 implies 3, Higasikawa [1995].
    58. 20 + 75 implies 359, Higasikawa [1995].
    We have also made additional changes to rfb1.tex (Part V). (After down loading the most recent version of rfb1.tex, run the program mk_book.c to get the updated versions of book1 and book2.)

  6. Add the bibliography item
    Howard, P./Keremedis, K./Rubin, J./Stanley, A./Tachtsis E.,
    [1999] Non-constructive properties of the real numbers, preprint.

  7. Add the two forms:
    FORM 92. C(WO,R): Every well ordered family of non-empty subsets of R has a choice function. Howard/Keremedis/Rubin/Stanley/Tachtsis [1999].
    FORM 337. C(WO,uniformly linearly ordered): If X is a well ordered collection of non-empty sets and there is a function f defined on X such that for every x in X, f(x) is a linear ordering of x, then there is a choice function for X. Howard/Keremedis/Rubin/Stanley/Tachtsis [1999].
    Both of these forms should be added to CHOICE FORMS, I. Choosing Single Elements in PART II.

  8. Add forms 34 and 92 to the list of forms that are transferable and to the list of forms that are true in all Fraenkel-Mostowski models. This should be done in notes 18 and 103 in Part IV, and in the introduction to Fraenkel-Mostowski models in Part III.

  9. There are several corrections and additions to NOTE 94.
    On page 279, line 17 from the bottom, change ``VII --> VI$''$'' to ``VII --> VII$''$''.
    On page 280, the implication at the bottom of the diagram, ``VII --> V$''$'' should be changed to ``VII --> VII$''$''
    The last paragraph starting with ``This leaves several problems open.'' Should be replaced by the following new results which are due to Omar de la Cruz. For this revision choose one of the two below:
  10. Several additions to rfb1.tex (PART V) have been made. (After down loading the most recent version of rfb1.tex, run the program mk_book.c to get the updated versions of book1 and book2.)

  11. Add note 39 as a reference for forms 89 and 217 in the Numerical List of Forms, Part I, and in the Topical List of Forms, Part II.

  12. The note number is missing from form [43 B] in both parts I and II. It should be note 126.